Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = csc²x − 2cot x, 0 < x < π

Verified step by step guidance

1. To find the local extrema of the function \( f(x) = \csc^2 x - 2 \cot x \) on the interval \( 0 < x < \pi \), we first need to find the derivative \( f'(x) \).
2. Recall that \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Use the chain rule and quotient rule to differentiate \( \csc^2 x \) and \( \cot x \).
3. The derivative of \( \csc^2 x \) is \( -2 \csc^2 x \cot x \), and the derivative of \( -2 \cot x \) is \( 2 \csc^2 x \). Therefore, \( f'(x) = -2 \csc^2 x \cot x + 2 \csc^2 x \).
4. Set \( f'(x) = 0 \) to find critical points: \( -2 \csc^2 x \cot x + 2 \csc^2 x = 0 \). Simplify to find \( \cot x = 1 \).
5. Solve \( \cot x = 1 \) within the interval \( 0 < x < \pi \) to find the critical points. Evaluate \( f(x) \) at these critical points to determine the local extrema.

Key Concepts

Trigonometric Functions

Understanding the trigonometric functions involved, such as cosecant (csc) and cotangent (cot), is crucial. The cosecant function is the reciprocal of sine, and cotangent is the reciprocal of tangent. These functions have specific properties and behaviors, especially within the interval 0 < x < π, which affect the function's behavior and the location of extrema.

Derivatives and Critical Points

To find local extrema, we need to calculate the derivative of the function and identify critical points where the derivative is zero or undefined. These points are potential candidates for local maxima or minima. Analyzing the derivative helps determine the function's increasing or decreasing behavior, which is essential for locating extrema.

Interval Analysis

The given interval, 0 < x < π, restricts the domain of the function, affecting where extrema can occur. It's important to consider the behavior of the function at the boundaries and within this interval. Since trigonometric functions can have discontinuities or undefined points, careful analysis within the specified range is necessary to accurately identify local extrema.